## Abstract

We establish a strong regularity property for the distributions of the random sums ∑ ± λ^{n}, known as "infinite Bernoulli convolutions": For a.e. λ ∈ (1/2, 1) and any fixed l, the conditional distribution of (ω_{n+1}, ω_{n+l}) given the sum ∑_{n=0} ^{∞}= ω_{n}λ^{n}, tends to the uniform distribution on (±1)^{l} as n → ∞. More precise results, where l grows linearly in n, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system of entropy h has K-partitions of any prescribed conditional entropy in [0, h]. This answers a question of Rokhlin and Sinai from the 1960's, for the case of Bernoulli systems.

Original language | American English |
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Pages (from-to) | 337-367 |

Number of pages | 31 |

Journal | Journal d'Analyse Mathematique |

Volume | 87 |

DOIs | |

State | Published - 2002 |

Externally published | Yes |